Group #15689049 — subgroups

Subgroups — #15689049

Order 3008.

170 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₄₇)⋊C₄	C₂	1	2	yes	maximal
2	C₂×(C₂×D₄₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#95	C₂	1	2	yes	maximal, complemented
4	C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₇	C₂²	1	2	yes	—
10	C₂²×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₄₇	—	2	2	no	—
12	(C₂×D₄₇)⋊C₄	—	2	4	no	—
13	(C₂×D₄₇)⋊C₄	—	2	2	no	—
14	C₂³×C₄₇	C₂³	1	1	yes	characteristic
15	C₂²×D₄₇	—	2	1	no	—
16	C₂²×D₄₇	—	2	2	no	—
17	C₂²×D₄₇	—	2	1	no	—
18	C₂²×D₄₇	—	2	2	no	—
19	C₂²×D₄₇	D₄	1	2	yes	—
20	C₂×C₄×C₄₇	—	2	2	no	—
21	C₂×C₄×C₄₇	—	2	2	no	—
22	C₂×C₄×C₄₇	—	2	2	no	—
23	C₂×C₄×C₄₇	—	2	2	no	—
24	C₂×C₄×C₄₇	D₄	1	2	yes	—
25	C₂×Dic₄₇	—	2	2	no	—
26	C₂×Dic₄₇	—	2	1	no	—
27	C₂×Dic₄₇	—	2	2	no	—
28	C₂×Dic₄₇	—	2	2	no	—
29	C₂×Dic₄₇	—	2	2	no	—
30	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄₇	#41	1	2	yes	—
33	C₂²×C₄₇	#41	1	1	yes	characteristic
34	C₂²×C₄₇	#41	1	2	yes	—
35	C₂×D₄₇	—	2	2	no	—
36	C₂×D₄₇	—	2	4	no	—
37	C₂×D₄₇	—	4	1	no	—
38	C₂×D₄₇	—	4	1	no	—
39	C₂×D₄₇	—	4	2	no	—
40	C₄×C₄₇	—	2	2	no	—
41	C₄×C₄₇	—	4	2	no	—
42	Dic₄₇	—	4	2	no	—
43	Dic₄₇	—	4	1	no	—
44	C₂×C₄₇	#121	1	1	yes	characteristic
45	C₂×C₄₇	#123	1	1	yes	characteristic
46	C₂×C₄₇	#123	1	2	yes	—
47	C₂×C₄₇	#124	1	1	yes	characteristic
48	C₂×C₄₇	#126	1	2	yes	—
49	D₄₇	—	4	2	no	—
50	#397	—	47	1	no	maximal, Sylow
51	C₄₇	#397	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₄⋊C₄	D₄₇	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	47	2	no	—
54	C₂×C₂²⋊C₄	—	47	1	no	—
55	#95	—	47	2	no	—
56	#95	—	47	1	no	—
57	C₂⁴	—	47	1	no	—
58	C₂²×C₄	—	47	2	no	—
59	C₂²×C₄	—	47	1	no	—
60	C₂²×C₄	C₂×D₄₇	1	2	yes	—
61	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	94	4	no	—
64	C₂²⋊C₄	—	94	2	no	—
65	C₂³	—	47	2	no	—
66	C₂³	—	94	2	no	—
67	C₂³	—	94	1	no	—
68	C₂³	—	94	2	no	—
69	C₂³	—	94	1	no	—
70	C₂³	C₂²×D₄₇	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	94	2	no	—
76	C₂×C₄	—	94	2	no	—
77	C₂×C₄	—	94	2	no	—
78	C₂×C₄	—	94	2	no	—
79	C₂×C₄	—	94	1	no	—
80	C₂×C₄	C₂²⋊D₄₇	1	2	yes	—
81	C₂²	—	94	2	no	—
82	C₂²	—	94	4	no	—
83	C₂²	—	188	1	no	—
84	C₂²	—	188	1	no	—
85	C₂²	—	188	2	no	—
86	C₂²	#10625350	1	2	yes	—
87	C₂²	D₄×D₄₇	1	1	yes	characteristic
88	C₂²	#10625352	1	1	yes	characteristic
89	C₂²	#10625354	1	2	yes	—
90	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	4	2	no	—
93	C₄	—	188	2	no	—
94	C₄	—	188	1	no	—
95	C₂	—	188	2	no	—
96	C₂	#13093995	1	1	yes	characteristic
97	C₂	#13094007	1	1	yes	characteristic
98	C₂	#13094009	1	2	yes	—
99	C₂	#13094013	1	1	yes	characteristic
100	C₂	#13094019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×(C₂×D₄₇)⋊C₄

C₂

yes

maximal

C₂×(C₂×D₄₇)⋊C₄

C₂

yes

maximal, characteristic

C₄₇⋊#95

C₂

yes

maximal, complemented

C₄₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂²

yes

characteristic

C₂²×Dic₄₇

C₂²

yes

—

C₂²×Dic₄₇

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₂³×C₄₇

C₂³

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

D₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄₇

#41

yes

—

C₂²×C₄₇

#41

yes

characteristic

C₂²×C₄₇

#41

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#124

yes

characteristic

C₂×C₄₇

#126

yes

—

D₄₇

—

#397

—

maximal, Sylow

C₄₇

#397

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

D₄₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

#95

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

—

C₂²×C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₄₇

yes

—

C₂²

—

C₂²

—

C₂²

—

188

—

C₂²

—

188

—

C₂²

—

188

—

C₂²

#10625350

yes

—

C₂²

D₄×D₄₇

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×C₂²⋊D₄₇

yes

characteristic

C₄

—

C₄

—

C₄

—

188

—

C₄

—

188

—

C₂

—

188

—

C₂

yes

C₂

yes

C₂

yes

—

C₂

#13094013

yes

characteristic

100

C₂

#13094019

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₇
commutator, socle, cosocle	C₂³×C₄₇	C₂³
Fitting	C₂×C₄⋊C₄×C₄₇	C₂
prime radical	C₂³×D₄₇	C₂²
hypercenter	C₂×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#397
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₄₇

commutator, socle, cosocle

C₂³×C₄₇

C₂³

Fitting

C₂×C₄⋊C₄×C₄₇

C₂

prime radical

C₂³×D₄₇

C₂²

hypercenter

C₂×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. C₄₇⋊#95	C₂	semidirect
4. C₄₇⋊#95	C₂	semidirect
5. C₂×C₄⋊C₄×C₄₇	C₂	semidirect
51. C₄₇	#397	semidirect
52. C₂×C₄⋊C₄	D₄₇	semidirect

H ≅ G/N

type

3. C₄₇⋊#95

C₂

semidirect

4. C₄₇⋊#95

C₂

semidirect

5. C₂×C₄⋊C₄×C₄₇

C₂

semidirect

51. C₄₇

#397

semidirect

52. C₂×C₄⋊C₄

D₄₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×(C₂×D₄₇)⋊C₄	2
C₄₇⋊#95	2
C₂×C₄⋊C₄×C₄₇	2
#397	47

subgroup

index

C₂×(C₂×D₄₇)⋊C₄

C₄₇⋊#95

C₂×C₄⋊C₄×C₄₇

#397

Subgroups — #15689049

Conjugacy classes of subgroups

Complements

Maximal subgroups